SMOOTHING OF DIGITAL PHOTOELASTIC DATA USING ROBUST OUTLIER
ALGORITHM
M. Ramji, K. Ramesh
Department of Applied Mechanics, Indian Institute of Technology Madras,
Chennai 600 036, India.
[email protected], [email protected]
ABSTRACT
With the advent of digital image processing systems, a separate branch of experimental technique namely digital
photoelasticity came into existence. In digital photoelasticity one can get photoelastic data of isoclinics and isochromatics at
every pixel in the domain. Unlike other interferometric techniques in digital photoelasticity one gets these photoelastic data as
two interdependent phase maps. This mutual dependence and interactions affect the photoelastic parameter evaluation.
Mathematically isoclinic values are undefined at isochromatic skeletons and this is termed as isochromatic-isoclinic interaction.
Due to isochromatic-isoclinic interaction noise is observed at these zones. Further isochromatic-isoclinic interaction increase
with increased load and this also adversely affect the isoclinic values. Thus a need for interpolating correct isoclinic values at
these zones is required. A few approaches for noise removal at these zones are compared and the suitability of the outlier
algorithm for such applications is demonstrated. The benchmark problem of a ring under diametral compression is used for the
purpose.
Introduction
Photoelasticity is an optical technique for experimental stress analysis. It is widely used for 2-D and 3-D analysis of
components for getting the information of principal stress difference (isochromatics) and principal stress direction (isoclinics) at
every point in the domain. With the advent of digital computers, recording of images as intensity data became easier and a
separate branch of photoelasticity namely digital photoelasticity came into existence [1]. In digital photoelasticity, intensity
information of the captured image is used for evaluating the isoclinic and isochromatic parameters. Thus, in principle one gets
values of isoclinic (θ) and isochromatic (N) for the whole-field in the form of phase map [1]. Phase shifting / polarization
stepping techniques are widely used in digital photoelasticity for getting the phase maps [1]. Among all the phase shifting
techniques (PST), six-step method stands out as it effectively accounts for the quarter-wave plate mismatch error [1, 2] and
background intensity [3]. Recently, Ramji et al. [4] have shown that plane polariscope based algorithms give better isoclinic
values than the methods that use a quarter wave plate.
The isoclinics and isochromatics are obtained only in a wrapped form directly from phase shifting / polarization stepping
algorithm. The wrapped values are represented as an image called phase map, which are different from conventional fringe
patterns of photoelasticity. They have to be unwrapped in different form for getting continuous phase values. Both isoclinic and
isochromatic phase maps have interdependence and this interaction affect their evaluation. Especially, the isoclinic values are
affected severely [5]. Mathematically, isoclinic values are undefined at isochromatic skeletons [2]. This is termed as
isochromatic-isoclinic interaction in the case of isoclinic phase map. The isochromatic-isoclinic interaction increases with
increased load and this is a major source of error in evaluating the isoclinic parameter [5]. Multiple wavelength approach [2],
multiple load approach [2], and interpolation [6] have been used in reducing noise occurring in isoclinic phase map due to
these interactions. Petrucci [7] has proposed a modified median filter for removing the noise appearing in the isoclinic data due
to electronics and irregularity of the surface of the specimen. One could also reduce the noise due to isochromatic-isoclinic
interaction by developing appropriate methods for data smoothing. Recently, Ramji and Ramesh [5] proposed a polynomial
based smoothing of isoclinic data. But smoothing has been done only along the horizontal direction. The polynomial fitting is a
global smoothing technique, which in principle can alter the trend of experimental isoclinic data in some zones. For stress
separation studies, one requires both isochromatics and isoclinics accurately free of any kinks in the domain.
In this paper, a comparative study of whole field smoothing algorithms for handling photoelastic data is done. Smoothing of
isoclinic data is done using median filter, least squares polynomial fitting and outlier algorithm. The problem of a ring under
diametral compression subjected to a moderately high load is used to compare the effectiveness of various smoothing
algorithms.
Photoelastic Data Acquisition
The isoclinic values are obtained using the plane polariscope based algorithm of Brown & Sullivan [8] and Table 1 summarises
the optical arrangements needed. The isoclinic value is obtained by
1
2
 I4 − I2 

 I 3 − I1 
θc = tan −1 
(1)
In Eq. (1), the subscript c indicates that the principal value of the inverse trigonometric function is used. Reference [1]
recommends that θc is to be evaluated by atan2 () function. The isoclinic value obtained by Eq. (1) is unwrapped and then
smoothed as discussed in the later part of this paper. These unwrapped isoclinic values (θ) are also used for isochromatic
phase map evaluation (Eq. (2)). The experimental data to evaluate isochromatics is extracted using a six-step PST algorithm
[2] based on a circular polariscope arrangement (Table 2). The isochromatic value is obtained by
 ( I 5 − I 3 ) sin 2θ + ( I 4 − I 6 ) cos 2θ 

( I1 − I 2 )


δ c = tan −1 
(2)
Again Ref. [1] recommends that δc is to be evaluated by atan2 () function. The phase shifted images are experimentally
recorded with a monochromatic light source (Sodium vapour, λ = 589.3 nm) and the intensity of light transmitted is recorded by
a monochrome CCD camera (DC-700 SONY) having a resolution of 768 × 576 pixels. Phase shifted images to be used in Eqs.
(1) & (2) are obtained one after another using the same polariscope and the recording system.
Table 1. Plane polariscope based algorithm [8] for isoclinic evaluation
α
β
π/2
0
5π/8
π/8
3π/4
π/4
7π/8
3π/8
Intensity Equation
I1 = I b + I a sin 2
δ
sin 2 2θ
2
Ia 2 δ
I 2 = I b + sin 1 − sin 4θ 
2
2
δ
I 3 = I b + I a sin 2 cos 2 2θ
2
Ia 2 δ
I 4 = I b + sin 1 + sin 4θ 
2
2
Table 2. Circular polariscope based algorithm [2] for isochromatic evaluation
ξ
η
β
3π/4
π/4
π/2
3π/4
π/4
0
I2
3π/4
0
0
I3
3π/4
π/4
π/4
I4
π/4
0
0
I5
π/4
3π/4
π/4
I6
Intensity Equation
Ia
(1 + cos δ )
2
I
= I b + a (1 − cos δ )
2
Ia
= I b + (1 − sin 2θ sin δ )
2
I
= I b + a (1 + cos 2θ sin δ )
2
I
= I b + a (1 + sin 2θ sin δ )
2
Ia
= I b + (1 − cos 2θ sin δ )
2
I1 = I b +
Isoclinic Phase Unwrapping
The procedure of phase unwrapping is explained by applying it to the benchmark problem of a ring under diametral
compression (outer dia = 80mm, inner dia = 40 mm, thickness = 6 mm, Fsigma = 11.54 N/mm/fringe). The polarization
stepping algorithm of Brown & Sullivan yields isoclinic values in the form of wrapped phase map. This wrapped isoclinic phase
map has to be unwrapped for getting continuous values. A new tile based quality guided phase unwrapping algorithm is used
for unwrapping the isoclinic phase map [9]. The tile chosen need not be only of rectangular shape but can be of any arbitrary
shape. This has been possible with the adaptation of a new approach for boundary encoding [10].
Unwrapped Isoclinic Phase Estimation
Experimentally generated isoclinic phase map for the problem of a ring under diametral compression (load = 503 N) is shown
in Fig. 1a. Figure 1e shows the theoretically obtained isoclinic plot [11]. By comparing Fig. 1a with Fig. 1e, one could see that
the isoclinic data correspond to different principal stress near the loading points both at the top & bottom and they are also at
left & right side of the model. These are termed as the inconsistent zones. In these zones, there is an abrupt variation in grey
scale because of the π/2 jump in isoclinic value. To correct these zones, quality based unwrapping algorithm developed by
Ramji and Ramesh is used [9]. Further, one could also see the presence of ninety degree jump along horizontal axis of
symmetry in Fig. 1e whereas it is not visible in Fig. 1a. Interestingly, the isoclinic data in the incorrect zone seems to be
continuous across the central line (Fig. 1a) and thereby making the unwrapping process more complicated. For a successful
unwrapping process, one needs to know before hand where these demarcations lie. The tiles for unwrapping needs to be
suitably selected based on it. Therefore, an additional image of the wrapped theta is derived with a different scaling where
white shade depicts +π/4 and black shade corresponds to -π/4 (Fig. 1b). This new figure is very useful in identifying the
demarcations mentioned above. On Fig. 1b, arbitrary boundaries are drawn using basic geometric primitives like line, circle,
arc, rectangle, Bezier curve, polygon etc. [10] such that it encloses both the correct and inconsistent zone. Especially tiles 3, 4,
7 and 8 are drawn considering the demarcation along the horizontal axis. After drawing various tiles, each inconsistent zone is
corrected individually.
Figure 1c shows the grey level representation of the unwrapped isoclinic phase map obtained after unwrapping by quality
guided algorithm. One could see a semblance of isochromatic fringes in the phase map. This is actually noise due to
isochromatic-isoclinic interaction. Figures 1d & 1f show a binary representation of isoclinic plot in steps of 10° obtained
experimentally and analytically. The binary representation (Fig. 1d) has clearly brought out more forcefully the bad impact of
noise due to isochromatic – isoclinic interaction on the isoclinic phase map. Thus, smoothing of raw unwrapped isoclinic data
is needed for further applications like stress separation studies.
Isoclinic Data Smoothing
For smoothing, the information of isoclinic data row by row within the model domain is required. Usually the phase data is
stored as an array and from this the retrieval of sequence of pixel coordinate values row by row within the model boundary is
required for implementing the smoothing algorithm. The effectiveness of different smoothing algorithms being developed such
as median filter, polynomial smoothing and robust outlier are compared both qualitatively and quantitatively.
Data Extraction and Management
Before doing the smoothing operation one must extract the data of the model domain. In digital domain, boundary is
represented as a set of pixel coordinates and to use this in applications such as smoothing it has to be encoded in a suitable
format. The first step is to initially identify the model boundary on a bright field image and draw them using the drawing
primitives such as line, circle, arc, Bezier curve, polygon etc. For retrieving boundary information, two file formats namely xbn
and ybn formats are defined. For example, consider the case of a specimen being circular. The xbn file format contains two ycoordinates, y1 and y2 corresponding to each vertical line on the image. Similarly ybn file format contains two x-coordinates,
x1 and x2 corresponding to each horizontal line on the image. The xbn file format is suited for vertical scanning algorithms and
ybn is suited for horizontal scanning algorithms. A software is developed using VC++ which has the provision of selecting
basic geometric primitives for boundary identification. An elegant algorithm is developed to extract xbn and ybn data of a
closed boundary drawn using the primitives by just selecting a point inside the domain [10].
As smoothing of the data is to be done row by row, one needs the co-ordinates of the boundary pixels as well as the number of
pixels in between them. These are extracted from the boundary information file with extension *.ybn. If needed, vertical
smoothing of the data is done column by column for which *.xbn file is used for extracting boundary information.
Smoothing by Median Filter
Median filter is predominantly used in electronic speckle pattern interferometry where the noise is randomly spread over the
model. This belongs to a class of spatial filters. Median filter is applied on the data directly. Initially filtering is done horizontally
row-wise and later applied vertically column-wise. The methodology of smoothing horizontally and then vertically is suitable for
multiply connected bodies. Median filtering consists of replacing each value by the median of the values in a window centered
about the pixel considered. When the filter is operating near the boundaries, the window spreads outside the model domain
and their values are to be suitably considered for evaluating the median. In this work, outside the boundaries the data value is
made zero before operating the filter. One has to specify the size of the window of the median filter and is problem dependent.
The algorithm is written in Matlab software.
π/2
π/4
3π/8
3π/16
2
1
π/4
π/8
π/8
8
3
7
4
0
0
-π/8
-π/4
(a)
-3π/8
π/16
-π/16
-π/8
(b)
5
6
-π/2
-3π/16
-π/4
π/2
3π/8
π/4
π/8
0
-π/8
-π/4
(c)
-3π/8
(d)
-π/2
π/2
3π/8
π/4
π/8
0°/ 90°
0
80°
60°40°
30°
20°
10°
80°
60°
40°
20°
10°
0°
-π/8
-π/4
-3π/8
(e)
-π/2
(f)
Figure 1 Steps involved in unwrapping isoclinic phase map obtained for the problem of a ring under diametral compression
obtained using Brown & Sullivan phase shifting algorithm (a) wrapped isoclinic (b) intermediate phase map image (with
arbitrary tile for ambiguity removal) (c) unwrapped isoclinic phase map (d) isoclinic plots in steps of 10° for experimentally
obtained unwrapped isoclinic values (e) unwrapped isoclinic phase map obtained using analytical solution [13] (f) isoclinic
plots in steps of 10° obtained analytically
Smoothing by Least Squares Polynomial
Ramji and Ramesh proposed a least squares polynomial based smoothing for the isoclinic values obtained from phase shifting
technique [5].This comes under the class of global regression techniques. The smoothing is termed global as it fits the
polynomial based upon the entire data considered. Ramji and Ramesh [5] developed the smoothing algorithm only along the
horizontal direction doing row by row. In this work, polynomial smoothing is extended along the vertical direction also. In least
squares polynomial smoothing, a polynomial curve is obtained by least squares analysis and then the smoothed values are
calculated back using that curve. It is done row by row and if needed later column by column. The only requirement is that one
needs to give the order of the polynomial for curve fitting. Again the polynomial order is problem dependent. The algorithm is
written in Matlab software and the polynomial function is available in the curve fitting module of the Matlab.
Smoothing by Robust Outlier
The outlier smoothing algorithm belongs to a class of local regression techniques. The smoothing procedure is termed local
because each smoothed value is determined by the neighbouring data points defined within a span. The span defines a
window of neighbouring points to be included in the smoothing calculation for each data point. The larger the span, the
smoothed curve will follow the trend better. The data points lying outside the trend are omitted and a local curve fitting is done
by least squares analysis [12]. The omission of data points that are present outside the trend is done by a weighting process.
The weights are calculated based upon the statistical parameter of median absolute deviation (MAD) and scaled between 0-1.
Data points with weights nearer to one are those which are lying outside the data trend locally and with weights nearer to zero
are those that exactly follow the data trend. Thus the algorithm avoids the near unity weighted data points and appropriately
chooses the data points closer to zero for local least squares curve fitting. Likewise, the whole procedure is completed for the
rest of the line. An important factor here is the selection of the span width and the order of the polynomial for least squares
curve fitting. It varies from problem to problem but usually, the longer the span width it is better. The algorithm is written in
Matlab software and the smoothing function is available in the curve fitting module of the Matlab. More details regarding this
algorithm can be found in the Ref. [12].
Performance of the algorithm on experimental isoclinic data
Qualitative Comparison
Since the problem of a ring is a multiply connected body, smoothing algorithm cannot be directly used. Concept of domain
delimiting is evolved for easier handling of models like ring. Domain delimiting is a technique to sub-divide the problem of
interest as an assembly of simply connected zones. The ring is divided into four quarters of simply connected nature as shown
in Fig. 2. For each of these regions, all the three algorithms are applied row-wise at first and then column-wise. In all four
hundred & seventy two rows and four hundred & seventy columns are present inside the model domain.
1
2
4
3
Figure 2 Figure showing the ring model split into four simply connected regions
Figure 3 shows the isoclinic phase map in grey scale obtained by various smoothing algorithms. Figure 3a is the isoclinic
phase map obtained after smoothing using the median filter. Here a 1-D span of twenty one is used along both the scanning
directions. One can clearly notice that a semblance of isochromatic fringes seen in Fig. 1c is also seen here. Figure 3b shows
the phase map obtained by least squares polynomial smoothing technique. A ninth order polynomial is found to be sufficient
for curve fitting along both the scanning directions. Here the semblance of isochromatic fringes is not seen whereas the region
along the horizontal axis of symmetry on both sides of the model has been badly affected.
Figure 3c shows the smoothed isoclinic phase map obtained using the robust outlier algorithm. A span length of forty pixels
and a linear polynomial for the least squares curve fitting is used along the horizontal direction. For smoothing along the
vertical direction, a span length of thirty pixels was found to be sufficient. One can clearly notice that a semblance of
isochromatic fringes is not seen here. Moreover it has performed well along the horizontal axis of symmetry compared to other
methods.
The grey scale representation of the results appears similar and one is not able to see the subtle variation between the
different approaches. Figure 4 shows the isoclinics as a binary representation in steps of 10°. When a binary representation of
the results is presented as in Fig. 4 one can clearly see that the performance of outlier algorithm is definitely a shade better
than the other two algorithms.
Quantitative Comparison
The smoothed theta value obtained by various smoothing algorithm is compared with the theta calculated theoretically for a
line at y/R = 0.51 in Fig. 5. Figure 5a shows the smoothed theta value obtained by using the median filter. There are many
undulations in the theta values. Figure 5b shows the theta values obtained by the least squares polynomial smoothing
algorithm. Here the theta values are mostly smooth except at the centre where there are many undulations. But when
compared to the median filter, polynomial smoothing has performed better. Figure 5c shows the smoothed theta values
obtained by the robust outlier algorithm. One could see that the values are smooth over the entire line and they follow the
analytical values closely. From this qualitative analysis, it is clear that the robust outlier algorithm has performed better than
the other two smoothing algorithms.
π/2
3π/8
π/4
π/8
0
-π/8
-π/4
(a)
(b)
-3π/8
(c)
-π/2
Figure 3 Smoothed isoclinic phase map in grey scale representation obtained by various smoothing algorithms for the problem
of a ring under diametral compression (a) Median filter (b) Least Square polynomial fitting (c) Robust outlier algorithm
(a)
(b)
(c)
Figure 4 Smoothed isoclinic plots in steps of 10° obtained by various smoothing algorithms for the problem of a ring under
diametral compression (a) Median filter (b) Least Square polynomial fitting (c) Robust outlier algorithm
Isochromatic Phase unwrapping and Data Smoothing
Isochromatic phase map obtained experimentally for the problem of a ring under diametral compression directly using the sixstep algorithm is shown in Fig. 6a. One can observe from Fig. 6a that close to the loading points, the fringes are discontinuous,
which are ambiguous zones. The isochromatic phase map should be free of ambiguous zones before unwrapping. Corrected
isoclinic value obtained from the plane polariscope arrangement is used for isochromatic calculation in Eq. (2) which gives the
phase map as in Fig. 6b. This is totally free of ambiguous zones. Now the corrected isochromatic phase map should be
unwrapped for getting the continuous phase values. Quality guided unwrapping by the concept of domain masking developed
recently for isoclinics [13] is used here. Unwrapped isochromatic phase map obtained using the quality guided approach is
shown in Fig. 6c.
Isochromatic data smoothing is similar to that of the isoclinic data smoothing. As the outlier algorithm has performed better for
isoclinic smoothing only this is considered for isochromatic data smoothing. The only change is in the selection of the span
width and the smoothing polynomial. The domain is delimited into four similar regions as shown in Fig. 2. For each of these
regions, the outlier algorithm is applied row-wise and column-wise using a span length of fifteen pixels and a linear polynomial
for the least squares curve fitting. Figure 6d shows the Matlab plot of the smoothed isochromatic values.
Conclusions
A versatile smoothing methodology using the outlier algorithm has been proposed for isoclinics initially and later adopted for
isochromatic smoothing. The application of the smoothing procedure for a multiply connected body is demonstrated using the
problem of a ring under diametral compression. For handling multiply connected body, the concept of domain limiting is used
where the problem domain is sub-divided into an assembly of simply connected regions. A comparative study of various whole
field smoothing methodologies is done. Among all the methodologies, robust outlier algorithm has performed well. The outlier
smoothing methodology drastically reduces the noise in the isoclinic parameter due to isochromatic-isoclinic interaction thus
yielding smooth values comparable with theory. The smoothing algorithm is also applied to isochromatic data obtained from
60
y/R= 0.51
Theta [deg]
40
20
0
-20
Theta_Analytical
-40
Theta_Smooth Median
-60
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x/R
60
y/R= 0.51
Theta [deg]
40
20
0
-20
Theta_Analytical
-40
Theta_Smooth Polynomial
-60
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x/R
60
y/R= 0.51
Theta [deg]
40
20
0
-20
Theta_Analytical
-40
Theta_Smoother Outlier
-60
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x/R
Figure 5 Comparison of smoothed theta obtained using various smoothing algorithms with analytically obtained theta along a
line (y/R=0.51) for the problem of ring under diametral compression (a) Median Filter (b) Least square polynomial smoothing
(c) Robust outlier algorithm
(a)
(b)
(c)
(d)
Figure 6 Steps involved in unwrapping isochromatic phase map obtained for the problem of a ring under diametral
compression obtained using six step phase shifting algorithm (a) phase map obtained using θ calculated by the six step
algorithm (b) phase map obtained using unwrapped isoclinic value from Brown & Sullivan algorithm in Eq. (2) (c) Matlab plot of
unwrapped isochromatic phase map obtained using phase derivative variance quality guided path follower unwrapping
algorithm (d) Matlab plot of smoothed isochromatic phase map obtained using robust outlier algorithm
phase shifting methodology. The smoothing algorithm developed is of a generic nature and can be applied to other optical
methods where data smoothing is required.
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